Multi-stage probabilistic signal shaping

ABSTRACT

A shaping encoder capable of improving the performance of PCS in nonlinear optical channels by performing the shaping in two or more stages. In an example embodiment, a first stage employs a shaping code of a relatively short block length, which is typically beneficial for nonlinear optical channels but may cause a significant penalty in the energy efficiency. A second stage then employs a shaping code of a much larger block length, which significantly reduces or erases the penalty associated with the short block length of the first stage while providing an additional benefit of good performance in substantially linear optical channels. In at least some embodiments, the shaping encoder may have relatively low circuit-implementation complexity and/or relatively low cost and provide relatively high energy efficiency and relatively high shaping gain for a variety of optical channels, including but not limited to the legacy dispersion-managed fiber-optic links.

BACKGROUND Field

Various example embodiments relate to optical communication equipmentand, more specifically but not exclusively, to signal encoders anddecoders.

Description of the Related Art

This section introduces aspects that may help facilitate a betterunderstanding of the disclosure. Accordingly, the statements of thissection are to be read in this light and are not to be understood asadmissions about what is in the prior art or what is not in the priorart.

Probabilistic signal shaping can beneficially provide energy savingsoften referred to as the shaping gain. In a typical implementation ofprobabilistic signal shaping (e.g., probabilistic constellation shaping,PCS), constellation symbols of relatively large energy are transmittedless frequently than constellation symbols of relatively small energy.For a linear communication channel, the shaping gain can theoreticallyapproach 1.53 dB.

SUMMARY OF SOME SPECIFIC EMBODIMENTS

Disclosed herein are various embodiments of a shaping encoder capable ofimproving the performance of PCS in nonlinear optical channels byperforming the shaping in two or more stages. In an example embodiment,a first stage employs a shaping code of a relatively short block length,which is typically beneficial for nonlinear optical channels but maycause a significant penalty in the energy efficiency. A second stagethen employs a shaping code of a much larger block length, whichsignificantly reduces or erases the penalty associated with the shortblock length of the first stage while providing an additional benefit ofgood performance in substantially linear optical channels. As a result,various embodiments may beneficially be used for both linear andnonlinear optical channels.

In some embodiments, a shaping encoder may employ an electronic codebookthat enables encoding equivalent to that of a two-stage shaping encoderto be performed using a corresponding search-and-match operation.

In at least some embodiments, the disclosed shaping encoder(s) may haverelatively low circuit-implementation complexity and/or relatively lowcost and provide relatively high energy efficiency and relatively highshaping gain for a variety of optical channels, including but notlimited to the legacy dispersion-managed fiber-optic links.

According to an example embodiment, provided is an apparatus,comprising: a digital encoder having, at least, first and second digitalstages to produce a stream of symbols from a bitstream, the firstdigital stage being configured to separately encode segments of a samenumber of bits of the bitstream into first sequences of symbols suchthat each of the first sequences has a same first length and arespective total energy of the symbols therein lower than a threshold,the second digital stage being configured to encode blocks of the firstsequences into second sequences of symbols, each of the blocks having asame number of first sequences therein, each of the second sequenceshaving a same second length; and wherein a total energy of symbols ofany of the second sequences divided by the second length is smaller thanthe threshold divided by the first length.

According to another example embodiment, provided is an apparatuscomprising an optical data transmitter that comprises an optical frontend and a digital shaping encoder, the digital shaping encoder beingconfigured to: encode a bitstream into a stream of symbols of aconstellation; and cause the optical front end to produce a modulatedoptical signal carrying the stream of symbols; and wherein the digitalshaping encoder is configured to generate a symbol sequence for thestream in response to a segment of the bitstream, the symbol sequenceincluding an integer number of non-overlapping subsequences of a fixedlength, any one of the subsequences having a transmit energy notexceeding a first threshold energy, the symbol sequence having atransmit energy not exceeding a second threshold energy and beinggenerated such that some symbols of higher energy have a lowerprobability of being generated by the digital shaping encoder than othersymbols of lower energy, the first threshold energy being smaller thanan average energy of an unshaped sequence of the fixed length of thesymbols of the constellation, the second threshold energy being smallerthan the integer number of the first threshold energies, the integernumber being greater than one.

BRIEF DESCRIPTION OF THE DRAWINGS

Other aspects, features, and benefits of various disclosed embodimentswill become more fully apparent, by way of example, from the followingdetailed description and the accompanying drawings, in which:

FIG. 1 shows a block diagram of a communication system in which variousembodiments can be practiced;

FIG. 2 graphically illustrates certain characteristics of an examplemodulated optical signal that can be transmitted in the communicationsystem of FIG. 1 according to an embodiment;

FIGS. 3A-3B graphically illustrate example probability distributionsthat can be applied to an 8-ary Pulse-Amplitude-Modulation (8-PAM)constellation in the communication system of FIG. 1 in some embodiments;

FIG. 4 shows a table of kurtosis values for several modulation formatsthat can be used in the communication system of FIG. 1 in someembodiments;

FIG. 5 graphically illustrates an 8-PAM constellation that can be usedin the communication system of FIG. 1 according to an embodiment;

FIG. 6 shows a block diagram of a multistage shaping encoder that can beused in the communication system of FIG. 1 according to an embodiment;

FIG. 7 shows a block diagram of a multistage shaping encoder that can beused in the communication system of FIG. 1 according to an alternativeembodiment;

FIG. 8 shows a block diagram of a shaping encoder that can be used inthe communication system of FIG. 1 according to yet another embodiment;

FIGS. 9A-9D illustrate an example codebook-construction procedure thatcan be used to construct a shaping codebook for the shaping encoder ofFIG. 8 according to an embodiment;

FIG. 10 shows a block diagram of a shaping decoder that can be used inthe communication system of FIG. 1 according to an embodiment;

FIG. 11 graphically illustrates example signal-to-noise ratio (SNR)improvements that can be achieved in the communication system of FIG. 1according to an embodiment;

FIG. 12 shows a block diagram of an optical transmitter that can be usedin the communication system of FIG. 1 according to an embodiment; and

FIG. 13 shows a block diagram of an optical receiver that can be used inthe communication system of FIG. 1 according to an embodiment.

DETAILED DESCRIPTION

Herein, the generation and/or transmission of a symbol stream in whichvarious symbols appear with different probabilities even though thevarious input data segments encoded onto said symbol stream have aboutequal probabilities is referred to as probabilistic constellationshaping (PCS). Often, preferable types of PCS generate symbol streams inwhich higher energy symbols are less probable than lower energy symbols.In some embodiments, forward-error-correction (FEC) encoding may be usedin a manner that causes such energy shaping to be largely maintained inthe corresponding FEC-encoded symbol stream. In embodiments of coherentoptical fiber communication systems and optical data transmitters andreceivers thereof, the PCS can advantageously be used, e.g., (i) tolower degradations related to nonlinear optical effects, which are moreprominent at larger energies, (ii) to lower a required signal-to-noiseratio (SNR) for a given transmission distance, and/or (iii) to enablehigher information communication rates. Various embodiments may applythe PCS to various quadrature amplitude modulation (QAM) constellationsand/or other suitable symbol constellations.

An important benefit of probabilistic signal shaping is that the amountof shaping (e.g., specific characteristics of the corresponding shapingcode) can be selected to optimize a desired set of performancecharacteristics of any given communication channel. For example,depending on specified performance requirements, probabilistic signalshaping can be adjusted to achieve an optimal (e.g., the highest)spectral efficiency or an optimal (e.g., the highest) net bit-rate forany given transmission distance.

FIG. 1 shows a block diagram of a communication system 100 in whichvarious embodiments can be practiced. System 100 comprises an opticaldata transmitter 104 and an optical data receiver 108 that are coupledto one another by way of a communication link 106. In an exampleembodiment, communication link 106 can be implemented using one or morespans of optical fiber or fiber-optic cable.

System 100 carries out PCS using (i) an electronic encoder 110appropriately interfaced with an electrical-to-optical (E/O) converter(also sometimes referred-to as the optical transmitter front end) 140 attransmitter 104, and (ii) an optical-to-electrical (O/E) converter (alsosometimes referred-to as the optical receiver front end) 150appropriately interfaced with an electronic decoder 160 at receiver 108.

In an example embodiment, one or both of electronic encoder 110 andelectronic decoder 160 can be implemented using a respective digitalsignal processor (DSP) or a portion thereof.

Electronic encoder 110 operates to generate one or more electricalradio-frequency (RF) signals 138 in response to receiving input data102. In response to electrical RF signal(s) 138, the opticalmodulator(s) of the E/O converter 140 then generate(s) a correspondingmodulated optical signal 142 suitable for transmission over link 106 andhaving encoded thereon a segment of the stream of input data 102. In anexample embodiment, E/O converter 140 comprises an optical modulatorthat can be implemented as known in the pertinent art, e.g., using: (i)a laser configured to generate an optical carrier wave; (ii) one or moremodulator configured to generate modulated optical signal 142 bymodulating the optical carrier wave generated by the laser; and (iii)one or more driver circuits, e.g., analog electrical circuits,configured to electrically drive the modulator(s) using electrical RFsignal(s) 138, thereby causing the E/O converter 140 to generatemodulated optical signal 142. Depending on the embodiment, the opticalmodulator(s) used in E/O converter 140 can be implemented using one ormore optical IQ modulators, Mach-Zehnder modulators, amplitudemodulators, phase modulators, and/or intensity modulators. The opticalmodulators may also be nested to generate a modulated optical carrierwhose orthogonal polarization components are separately modulated. Insome embodiments, E/O converter 140 may employ directly modulatedlasers, e.g., laser diodes configured to generate modulated opticalsignals in response to modulated electrical currents that drive thediodes.

Communication link 106 typically imparts noise and other linear and/ornonlinear signal impairments onto signal 142 and delivers a resultingimpaired (e.g., noisier) signal 142′ to O/E converter 150 of receiver108. O/E converter 150 operates to convert optical signal 142′ into oneor more corresponding electrical RF signals 152. Electronic decoder 160then uses decoding processing to recover data 102 from the electrical RFsignal(s) 152.

In some embodiments, O/E converter 150 comprises an optical demodulatorthat can be configured, as known in the pertinent art, for coherent(e.g., intradyne or homodyne) detection of signal 142′. In suchembodiments, O/E converter 150 may include: (i) an opticallocal-oscillator (LO) source; (ii) an optical hybrid configured tooptically mix signal 142′ and the LO signal generated by the optical LOsource; and (iii) one or more photodetectors (e.g., photodiodes)configured to convert optical interference signals generated by theoptical hybrid into the corresponding components of electrical RFsignal(s) 152.

In some other embodiments, O/E converter 150 comprises an opticaldemodulator that can be configured for direct (e.g., square law,intensity, optical power) detection of signal 142′. In such embodiments,O/E converter 150 may include a photodiode configured to generateelectrical RF signal 152 to be proportional to the intensity (opticalpower, squared amplitude of the electric field) of signal 142′, i.e.,without mixing the received signal 142′ with light of a local opticalsource.

FIG. 2 graphically illustrates certain characteristics of an exampleoptical signal 142 that can be transmitted in communication system 100(FIG. 1 ) according to an embodiment. In this particular example,optical data transmitter 104 is configured to generate optical signal142 using a 64-QAM constellation 210, the constellation symbols of whichare shown as points on an IQ plane 202, wherein the I and Q coordinateaxes are the in-phase and quadrature-phase dimensions, respectively, ofthe IQ plane. In an example embodiments, the constellation-symbolamplitudes in constellation 210 in each of the I and Q dimensionsthereof can be represented by the integers −7, −5, −3, −1, +1, +3, +5,and +7.

A three-dimensional (3D) bar chart 220 also shown in FIG. 2 graphicallyillustrates relative probability of occurrence of the differentconstellation symbols of constellation 210 in optical signal 142. Inparticular, chart 220 indicates that electronic encoder 110 may operateto generate a constellation-symbol stream for being carried by opticalsignal 142 such that constellation symbols of a first transmit energyoccur with higher probability than constellation symbols of a secondtransmit energy that is greater than the first transmit energy.

FIGS. 3A-3B graphically illustrate example probability distributionsthat can be applied by electronic encoder 110 to the I or Q dimension ofconstellation 210 in some embodiments. Distribution 310 shown in FIG. 3Ais a Maxwell-Boltzmann (MB) distribution that can be expressed using Eq.(1):P∝exp(−λ·|x| ²)  (1)where P is the probability of a constellation symbol; x is the amplitudeof the constellation symbol in the I or Q dimension of the IQ plane 202;and λ is a positive real value. The value of λ is related to the rate H(e.g., entropy) of the shaping code used in electronic encoder 110 andcan be selected, e.g., based on a desired code-rate value. Distribution320 shown in FIG. 3B is a generalized Maxwell-Boltzmann (GMB)distribution that can be expressed using Eq. (2):P∝exp(−λ˜|x| ^(α))  (2)where the parameter a is allowed to have any positive real value. The MBdistribution is a special case of the GMB distribution, wherein α=2. Inthe GMB example shown in FIG. 3B, α=3.5. As the value of a increasesabove α=2, the GMB distribution becomes closer to a uniformdistribution, e.g., as visually evident from a comparison ofdistributions 310 and 320 and the flatter middle portion of the latter.For the same shaping-code rate H, distribution 320 typically results ina larger average energy of optical signal 142 than distribution 310. Forexample, for H=2.6 bit/symbol, the average-energy difference betweendistributions 310 and 320 is approximately 0.15 dB.

Distributions 310 and 320 are also characterized by different respectivevalues of kurtosis (hereafter denoted as K). More specifically, forH=2.6, distribution 310 has K=0.44, whereas distribution 320 has K=0.31.These kurtosis values have been calculated using the absolute-valuedsignals that have a positive mean value (or, equivalently, using onlythe positive portion of the constellation). Kurtosis values calculatedwithout the absolute-value operator, e.g., for signals having the zeromean value, may be different.

In probability theory and statistics, kurtosis is a measure of the“tailedness” of the probability distribution of a real-valued variable.Kurtosis can be used, e.g., to quantify the shape of a probabilitydistribution. Herein, kurtosis refers to the standardized fourth-ordermoment of the probability distribution.

FIG. 4 shows a table of kurtosis values for several example modulationformats that can be used in system 100 in some embodiments. Therein, PSstands for “probabilistically shaped” and indicates the use of an MBdistribution. The shown kurtosis values have been calculated usingabsolute-valued signals, as mentioned above. Different MB distributionscorresponding to the different PS 16-QAM constellation entries in thetable of FIG. 4 correspond to different respective values of A, whichare manifested by the different respective values of the shaping-coderate H shown therein. The entries of the table of FIG. 4 indicate that,for the same constellation, e.g., 16-QAM, kurtosis can be a nonlinearfunction of the shaping-code rate H. The entries of the table furtherindicate that kurtosis may also depend on the constellation size.

For an additive white Gaussian noise (AWGN) channel, an optimaldistribution of QAM constellation symbols may be an MB distribution,such as the distribution 310 (FIG. 3A). However, for at least someoptical fiber channels, a nonlinear interference noise (NLIN) may alsobe present in addition to the AWGN. If optical-signal power increases,then the NLIN increases. If kurtosis increases, then the NLIN increases.As such, kurtosis of the employed modulation format may be used as anapproximate measure of the relative amount of NLIN in the correspondingoptical channel. However, it should be noted that, for a givenmodulation format, kurtosis does not vary with the optical-signal power.Both AWGN and NLIN detrimentally reduce the effective SNR at thereceiver.

As illustrated by the table of FIG. 4 , at least some MB distributionsmay have higher kurtosis values than the corresponding uniformdistribution(s). As a result, the use of MB distributions in somecommunication systems, e.g., systems employing highly nonlinear opticalfiber links, such as the legacy dispersion-managed links that aretypically operated with a relatively high optical launch power, mayimpose a significant SNR penalty on the received optical signal, e.g.,optical signal 142′ (see FIG. 1 ), due to the NLIN. At least some GMBdistributions may in some cases reduce the SNR penalty with respect tothat of the corresponding MB distribution(s), e.g., as indicated by theabove-mentioned kurtosis values corresponding to distributions 310 and320.

FIG. 5 graphically illustrates an 8-ary Pulse-Amplitude-Modulation(8-PAM) constellation 510 that can be used to program electronic encoder110 according to an embodiment. The eight constellation points ofconstellation 510 are all located on the I-axis of the I-Q plane. Eachof the constellation points is used to encode three bits. An example of3-bit bit-words assigned to different constellation points ofconstellation 510 is shown above the I-axis in FIG. 5 . The relativeamplitudes corresponding to the different constellation points ofconstellation 510 are shown below the I-axis in FIG. 5 and can berepresented by the integers −7, −5, −3, −1, +1, +3, +5, and +7.

In the illustrated embodiment, the constellation-point labeling is inaccordance with a reflected double-Gray mapping scheme, in which theconstellation points located in the positive I-half of constellation 510have binary-amplitude labels (i.e., the labels that do not include thesign bit) generated using conventional double-Gray mapping, while theconstellation points located in the negative I-half of the constellationhave binary labels generated by flipping the sign bits of thecorresponding constellation points located in the positive I-half. Withthis type of mapping, the amplitude labels of the constellation pointsare symmetric, and the sign bits of the constellation points areanti-symmetric with respect to the origin of the I-axis.

In some alternative embodiments, a similar approach can be used togenerate binary labels (i.e., assigned bit-words) for a constellation ora constellation portion that uses the Q dimension of the complex I-Qplane. For example, constellation-point labeling for the 64-QAMconstellation 210 (FIG. 2 ) can be implemented using the labeling shownin FIG. 5 , with such labeling being applied to both I and Q dimensionsof constellation 210.

For illustration purposes and without any implied limitations, thedescription of some example embodiments is given herein below inreference to a 4-PAM constellation. A 4-PAM constellation is similar toconstellation 510, but has four distinct constellation pointsdistributed along a 1-dimensional line. Herein, we assume that theconstellation points are arranged equidistantly with respect to eachother and symmetrically around the origin (zero) and can be representedby the integers −3, −1, 1, and 3. An extension of the presenteddescription to any 2^(m)-PAM constellation is relativelystraightforward, where m is an integer greater than one. Furtherextensions to other possible dimensions of optical signal 142, e.g.,time, polarization, spatial mode, and frequency/wavelength, will beapparent to a person of ordinary skill in the pertinent art without anyundue experimentation. For example, two 2^(m)-PAM symbols can becombined to construct a 2^(2m)-QAM symbol by modulating each of the Iand Q dimensions of the optical signal 142 independently with arespective 2^(m)-PAM symbol. A person of ordinary skill in the art willunderstand that, for example, the 64-QAM constellation 210 of FIG. 2corresponds to m=3.

As an illustrative example, let us consider a single-stage shapingencoder that produces a length-n symbol sequence X=[x₁ . . . x_(n)] inresponse to a length-k binary sequence B=[b₁ . . . b_(k)]. The totalenergy of the sequence X can be calculated as:

$\begin{matrix}{{X}^{2} = {\sum\limits_{i = 1}^{i = n}{❘x_{i}❘}^{2}}} & (3)\end{matrix}$For example, for 4-PAM, if n=8 and X=[1, 1, 1, 1, 1, 3, 3, 3], then Eq.(3) results in ∥x∥²=32 (=1+1+1+1+1+9+9+9). In general, such a shapingencoder can be configured to apply a shaping code that, for a selectedfixed n, can make the total energy ∥X∥² of the sequence X to be notlarger than a selected fixed threshold energy E_(max), i.e., anysequence X of length n at the output of the shaping encoder fulfills theinequality:∥X∥ ² ≤E _(max) <U _(n)  (4)where U_(n) is the average energy of the unshaped length-n symbolsequence X, i.e., a sequence in which different constellation symbolsoccur with an approximately equal probability. For the above example of4-PAM and n=8, said average energy is U_(n)=40.

Since Eq. (3) is analogous to the expression for the squared Euclideannorm of the vector X=(x₁ . . . x_(n)), wherein x₁, . . . , x_(n) are theCartesian coordinates of the vector in the n-dimensional space, it canbe said that all vectors X produced by such a shaping encoder areenclosed by the n-dimensional sphere of radius √{square root over(E_(max))} centered on the origin of the coordinate system. Such shapingcodes are sometimes referred to in the literature as “sphere shaping”codes. Representative examples of algorithms that can be adapted forimplementing sphere shaping are described, e.g., in (1) Patrick Schulte,Georg Böcherer, “Constant Composition Distribution Matching,” IEEETRANSACTIONS ON INFORMATION THEORY, 2016, vol. 62, No. 1, pp. 430-434;and (2) Yunus Can Gültekin, et al., “Approximate Enumerative SphereShaping,” 2018 IEEE International Symposium on Information Theory(ISIT), pp. 676-680, both of which are incorporated herein by referencein their entirety.

For a given constellation (e.g., 510, FIG. 5 ) and a fixed code rate H(=k/n bits per symbol), a sphere-shaping code produces a probabilitydistribution whose kurtosis depends on the block length n of the code.For example, a shorter-length (e.g., length-n₁) sphere-shaping codetypically produces a probability distribution with a smaller kurtosisthan a longer-length (e.g., length-n₂, n₂>n₁) sphere-shaping code.However, the shorter-length sphere-shaping code is typically lessenergy-efficient than the longer-length sphere-shaping code or acorresponding MB code. Circuit-implementation complexity typicallyincreases with the block length n of the sphere-shaping code.

It would be desirable to provide, e.g., for communication system 100, ashaping encoder that has at least some of the following characteristics:(i) has relatively low circuit-implementation complexity and/orrelatively low cost; (ii) results in relatively high energy efficiencyfor transmission of payload data; (iii) provides a relatively highshaping gain for NLIN-impaired optical channels while being also capableof providing good performance for substantially linear optical channels;(iv) lends itself to convenient interfacing with an FEC encoder; (v)enables optical-signal transmission over a relatively large distance,e.g., larger than the reach of the corresponding unshaped opticalsignal; and (iv) can support relatively high data throughput.

FIG. 6 shows a block diagram of a multistage shaping encoder 600 thatcan be used in electronic encoder 110 according to an embodiment.Shaping encoder 600 has a nested architecture that is illustrativelyshown in FIG. 6 as comprising L nested stages 602 ₁-602 _(L), where L isan integer greater than two. In some alternative embodiments, shapingencoder 600 may have only two stages, i.e., correspond to L=2. The firstencoder stage 602 ₁ comprises a shaping encoder 610 ₁. Each subsequentencoder stage 602 _(i) (where i=2, . . . , L) comprises the previousencoder stage 602 _(i-1) serially connected with a constellationdemapper 620 _(i) and a shaping encoder 610 _(i). For example, encoderstage 602 ₂ comprises encoder stage 602 ₁ serially connected withconstellation demapper 620 ₂ and shaping encoder 610 ₂, and so on.

For illustration purposes, multistage shaping encoder 600 is describedbelow as having been programmed to use a 2^(m)-PAM constellation, e.g.,constellation 510 (FIG. 5 ).

In an example embodiment, shaping encoder 610 ₁ operates to apply afirst sphere-shaping code to an input bitstream 608 ₁ to generate anoutput amplitude stream 612 ₁ of positive amplitudes. The firstsphere-shaping code has a rate H₁=k₁/n₁ and a threshold energy E₁ and isapplied to bitstream 608 ₁ in a segment-by-segment manner. Morespecifically, in response to a first received segment of k₁ bits ofbitstream 608 ₁, shaping encoder 610 ₁ generates a correspondingsequence of n₁ positive amplitudes for amplitude stream 612 ₁. Inresponse to a next non-overlapping segment of k₁ bits of bitstream 608₁, shaping encoder 610 ₁ generates a corresponding next sequence of n₁positive amplitudes for amplitude stream 612 ₁, which is appended to thepreviously outputted sequence of n₁ positive amplitudes, and so on.Amplitude stream 612 ₁ may include only the positive amplitudes of theoperative 2^(m)-PAM constellation. For example, if constellation 510(FIG. 5 ) is the operative constellation, then amplitude stream 612 ₁may include only the amplitudes 1, 3, 5, and 7 (see FIG. 5 ). If theoperative constellation is 4-PAM, then amplitude stream 612 ₁ mayinclude only the amplitudes 1 and 3. The first sphere-shaping codetypically causes a smaller amplitude to occur in amplitude stream 612 ₁with a higher relative probability than a larger amplitude.

Demapper 620 ₂ operates to convert amplitude stream 612 ₁ into acorresponding bitstream 608 ₂ using the amplitude bits of the binarylabels of the corresponding constellation points. In the example ofconstellation 510 (FIG. 5 ), the amplitude bits are the two leastsignificant bits (LSBs) of each binary label. In this case, eachamplitude 1 is converted by demapper 620 ₂ into the binary “11” forbitstream 608 ₂; each amplitude 3 is converted by demapper 620 ₂ intothe binary “10” for bitstream 608 ₂; each amplitude 5 is converted bydemapper 620 ₂ into the binary “01” for bitstream 608 ₂; and eachamplitude 7 is converted by demapper 620 ₂ into the binary “00” forbitstream 608 ₂.

Shaping encoder 610 ₂ operates to apply a second sphere-shaping code tobitstream 608 ₂ to generate an output amplitude stream 612 ₂ of positiveamplitudes. The second sphere-shaping code has a code rate H₂=k₂/n₂ anda threshold energy E₂ and is applied to bitstream 608 ₂ in asegment-by-segment manner. More specifically, in response to a firstsegment of k₂ bits of bitstream 608 ₂, shaping encoder 610 ₂ generates acorresponding sequence of n₂ positive amplitudes for amplitude stream612 ₂. In response to a next non-overlapping segment of k₂ bits ofbitstream 608 ₂, shaping encoder 610 ₂ generates a corresponding nextsequence of n₂ positive amplitudes for amplitude stream 612 ₂, which isappended to the previously outputted sequence of n₂ positive amplitudes,and so on. Similar to amplitude stream 612 ₁, amplitude stream 612 ₂ mayinclude only the positive amplitudes of the operative 2^(m)-PAMconstellation. The second sphere-shaping code typically causes a smalleramplitude to occur in amplitude stream 612 ₂ with a higher relativeprobability than a larger amplitude.

In some embodiments, the input block lengths for the first and secondsphere-shaping codes may be the same, i.e., k₁=k₂.

In an example embodiment, n₂>n₁ and n₂/n₁=ρ, where ρ is an integergreater than one. The threshold energy E₂ is selected such as to cause asegment of amplitude stream 612 ₂ to have a smaller transmit energy thanthat of an equal-length segment of amplitude stream 612 ₁, e.g.,according to the following inequality:E ₂ <ρE ₁  (5)A functional relationship of any next stage 602 _(i) with thecorresponding preceding stage 602 _(i-1) is generally analogous to theabove-described functional relationship between the stages 602 ₂ and 602₁.

Demapper 620 _(L) operates to convert amplitude stream 612 _(L-1)received form stage 604 _(L-1) into a corresponding bitstream 608 _(L)using the amplitude bits of the binary labels of the correspondingconstellation points. Shaping encoder 610 _(L) operates to apply an L-thsphere-shaping code to bitstream 608 _(L) to generate an outputamplitude stream 612 _(L) of positive amplitudes. The L-thsphere-shaping code has a rate H_(L)=k_(L)/n_(L) and a threshold energyE_(L) and is applied to bitstream 608 _(L) in a segment-by-segmentmanner. The threshold energy E_(L) is selected such as to cause asegment of amplitude stream 612 _(L) to have a smaller transmit energythan that of an equal-length segment of amplitude stream 612 _(L-1).

Shaping encoder 600 further comprises a multiplier 630 that operates totransform amplitude stream 612 _(L) into a corresponding amplitudestream 632 carrying signed amplitudes, i.e., positive and negativeamplitudes. The conversion is performed in response to a bitstream 628,in which binary “zeros” and “ones” have about equal probability ofoccurrence. In response to a binary “zero” received via bitstream 628,multiplier 630 multiplies the corresponding received amplitude ofamplitude stream 612 _(L) by −1, thereby outputting a negative amplitudefor amplitude stream 632. In response to a binary “one” received viabitstream 628, multiplier 630 multiplies the corresponding receivedamplitude of amplitude stream 612 _(L) by +1, thereby outputting apositive amplitude for amplitude stream 632. The resulting amplitudestream 632 can then be used to modulate the optical carrier, as alreadyindicated above.

In some embodiments, at least some segments of bitstream 628 may includeparity bits, e.g., generated by applying a suitable FEC code tobitstream 608 ₁. In some embodiments, bitstream 628 can be generatedusing some of the methods and/or circuits disclosed in U.S. Pat. Nos.10,727,951, 10,523,400, 10,200,231, and 10,091,046, all of which areincorporated herein by reference in their entirety.

In an example embodiment, appropriate selection of L sphere-shapingcodes for the L stages of shaping encoder 600 advantageously enablesamplitude stream 632 to have a probability distribution that has adesired value of kurtosis, e.g., a value that provides nearly optimalperformance for a given communication link 106. In particular, theprobability distribution of amplitude stream 632 can be adjusted, bychanging the code selection, to properly mitigate the adverse effects ofthe given amounts of AWGN and NLIN in link 106.

In some embodiments, shaping encoder 600 can beneficially be implementedusing a less-complex digital circuit than that of a comparablesingle-stage shaping encoder configured to realize a GMB distribution(see Eq. (2)).

FIG. 7 shows a block diagram of a multistage shaping encoder 700 thatcan be used in electronic encoder 110 according to an alternativeembodiment. Shaping encoder 700 has a serial architecture and isillustratively shown in FIG. 6 as comprising L serially connectedshaping encoders 610 ₁, 710 ₂-710 _(L), where L is an integer greaterthan two. In some alternative embodiments, multistage shaping encoder700 may have only two serially connected shaping encoders, i.e.,encoders 610 ₁, 710 ₂ for L=2.

Shaping encoder 610 ₁ is already described above in reference to FIG. 6. Each of shaping encoders 710 ₂-710 _(L) differs from shaping encoder610 ₁ in that a shaping encoder 710 receives an amplitude stream andoutputs another amplitude stream, whereas shaping encoder 610 ₁ receivesa bitstream and outputs an amplitude stream. For example, shapingencoder 710 ₂ operates to apply a second sphere-shaping code toamplitude stream 612 ₁ to generate an amplitude stream 712 ₂ of positiveamplitudes. The second sphere-shaping code has a fractional rateR₂=n₁/n₂<1 and a threshold energy E₂ and is applied to amplitude stream612 ₁ in a segment-by-segment manner. The threshold energy E₂ isselected such as to cause a segment of amplitude stream 712 ₂ to have asmaller transmit energy than that of an equal-length segment ofamplitude stream 612 ₁.

Shaping encoder 710 _(L) operates to apply an L-th sphere-shaping codeto amplitude stream 712 _(L-1) to generate an output amplitude stream712 _(L) of positive amplitudes. The L-th sphere-shaping code has afractional rate R_(L)=n_(L-1)/n_(L)<1 and a threshold energy E_(L) andis applied to amplitude stream 712 _(L-1) in a segment-by-segmentmanner. The threshold energy E_(L) is selected such as to cause asegment of amplitude stream 712 _(L) to have a smaller transmit energythan that of an equal-length segment of amplitude stream 712 _(L-1).That is,

$\frac{E_{L}}{n_{L}} < {\frac{E_{L - 1}}{n_{L - 1}}.}$

Shaping encoder 700 further comprises a multiplier 630 that operates totransform amplitude stream 712 _(L) into a corresponding amplitudestream 732 carrying signed amplitudes. The conversion is performed inresponse to bitstream 628, e.g., as already described above in referenceto FIG. 6 .

FIG. 8 shows a block diagram of a shaping encoder 800 that can be usedin electronic encoder 110 according to yet another embodiment. Shapingencoder 800 implements the functionality of shaping encoder 700 using asegment matcher 810 and a codebook store 820. In operation, segmentmatcher 810 receives a length-k₁ segment of bitstream 608 ₁, finds amatch to this segment in codebook store 820, retrieves a correspondinglength-n_(L) amplitude sequence from the codebook store, and outputs theretrieved amplitude sequence to generate a length-n_(L) segment foramplitude stream 712 _(L).

Codebook store 820 has stored therein a shaping codebook constructed tocompress the multistage encoding of shaping encoder 700 into a singlesearch-and-match operation, but otherwise produces the same amplitudestream 712 _(L) as shaping encoder 700 in response to the same bitstream608 ₁. In an example embodiment, a shaping codebook for codebook store820 can be constructed, e.g., as described below in reference to anexample shaping codebook 900 corresponding to a 4-PAM constellation(i.e., m=2), L=2, k₁=36, n₁=4, n₂=60, E₁=20, E₂=124, H=k₁/n₂=0.6bit/amplitude. A person of ordinary skill in the pertinent art will beable to use a similar codebook-construction procedure to construct othershaping codebooks corresponding to other sets of the above-listedparameters.

FIGS. 9A-9D illustrate an example codebook-construction procedure thatcan be used to construct shaping codebook 900 for codebook store 820according to an embodiment.

FIG. 9A shows all possible length-4 amplitude sequences corresponding tom=2.

FIG. 9B illustrates a codebook-construction step in which the amplitudesequences of FIG. 9A are sorted and pruned to remove the sequences thatdo not satisfy the E₁=20 energy constraint, i.e., to remove the length-4sequences for which ∥X∥²>20. The remaining sequences form a sequence set910, which has N=11 different length-4 amplitude sequences.

FIG. 9C illustrates a codebook-construction step in which all possiblecombinations of amplitude sequences from the set 910 (FIG. 9B) of length60 are enumerated. The total number of such combinations is 11¹⁵=N^(C),where C=n₂/n₁=15. In the decimal format, 11¹⁵ 4.18×10¹⁵. This number ofamplitude sequences can be used to encode └log₂11¹⁵┘=51 bits.

FIG. 9D illustrates a codebook-construction step in which the amplitudesequences of FIG. 9C are sorted and pruned to remove the amplitudesequences that do not satisfy the E₂=124 energy constraint, i.e., toremove the length-60 amplitude sequences for which ∥X∥²>124. Some morelength-60 amplitude sequences for which ∥X∥²=124 may need to be removedfrom the sorted list of amplitude sequences to limit the remainingnumber of amplitude sequences in a set 920 exactly to 2³⁶≈6.87×10¹⁰.These remaining amplitude sequences can then be mapped, one-to-one, to2³⁶ different length-36 bit sequences in any suitable manner. Theresulting mapping of 2³⁶ different length-36 bit sequences to 2³⁶different length-60 amplitude sequences can serve as shaping codebook900, which can be loaded into the codebook store 820 (FIG. 8 ) and usedas described above.

FIG. 10 shows a block diagram of a shaping decoder 1000 that can be usedin electronic decoder 160 according to an embodiment. Shaping decoder1000 is compatible with shaping encoders 700 and 800 and performsamplitude-sequence decoding that is inverse to the encoding performed byshaping encoders 700 and 800. For example, in response to amplitudestream 732 received from the upstream circuitry of receiver 108, shapingdecoder 1000 operates to recover bitstream 608 ₁, as indicated in FIG.10 .

Shaping decoder 1000 comprises a demultiplexer 1030, a sequence matcher1010, and a copy of codebook store 820. Demultiplexer 1030 operates todemultiplex amplitude stream 732 (which carries signed amplitudes, asexplained above in reference to FIGS. 7, 8 ) into bitstream 628 (whichcarries sign bits of the signed amplitudes) and amplitude stream 712_(L) (which carries unsigned amplitudes). Sequence matcher 1010 operatesto receive a length-n_(L) amplitude sequence of amplitude stream 712_(L), finds a match to this sequence in codebook store 820, retrieves acorresponding length-k₁ bit-word from the codebook store, and outputsthe retrieved bit-word to generate a length-k₁ segment for bitstream 608₁.

The codebook store 820 of shaping decoder 1000 has therein the sameshaping codebook as that of the codebook store 820 of the counterpartshaping encoder 800. For example, the codebook store 820 of shapingdecoder 1000 may be loaded with a copy of the above-described shapingcodebook 900. A person of ordinary skill in the art will understandthat, in some embodiments, shaping decoder 1000 may be adapted in arelatively straightforward manner to perform decoding that is inverse tothe encoding of shaping encoder 700.

FIG. 11 graphically illustrates example SNR improvements that can beachieved according to an embodiment. The SNR data shown in FIG. 11represent computer simulations of a representative legacydispersion-managed communication link 106. The optical launch power of−7 dBm corresponds to an approximately optimum launch power of theunshaped QPSK-modulated optical signal, which is typical of the legacydesign target. A curve 1102 graphically shows SNR results correspondingto a 1-stage shaping algorithm implemented using the ConstantComposition Distribution Matching (CCDM) described in the above-citedSchulte reference, with the shaping-code rate of H=0.6, the amplitudesequence lengths n=5,120, and the CCDM being configured to implement theMB distribution for a 4-PAM constellation. A curve 1104 graphicallyshows SNR results corresponding to a 1-stage shaping algorithmimplemented using the Enumerative Sphere Shaping (ESS) described in theabove-cited Gültekin reference, with the shaping-code rate of H=0.6 andthe amplitude sequence lengths n=60 for the same 4-PAM constellation. Acurve 1106 graphically shows SNR performance data corresponding to a2-stage shaping (2-SS) algorithm implemented using a shaping codebookanalogous to the above-described shaping codebook 900 but constructedfor n₂=20. The higher SNR values of curve 1106 attest to the betterperformance of the 2-stage shaping algorithm with respect to either ofthe above-mentioned 1-stage shaping algorithms. For example, at theoptical launch power of −7 dBm and transmission distance of 2400 km, the2-stage sphere-shaping algorithm achieves ˜1.5-dB gain in effective SNRrelative to the MB CCDM shaping algorithm.

FIG. 12 shows a block diagram of optical transmitter 104 that can beused in system 100 (FIG. 1 ) according to an embodiment.

In operation, transmitter 104 receives input stream 102 of payload dataand applies it to a digital signal processor (DSP) 112, whichimplements, inter alia, the electronic encoder 110 (FIG. 1 ). DSP 112processes input data stream 102 to generate digital signals 114 ₁-114 ₄.In an example embodiment, DSP 112 may perform, one or more of thefollowing: (i) de-multiplex input stream 102 into two sub-streams, eachintended for optical transmission using a respective one of orthogonal(e.g., X and Y) polarizations of optical output signal 142; (ii) encodeeach of the sub-streams using one or more suitable codes, e.g., asoutlined above; and (iii) convert each of the two resulting sub-streamsinto a corresponding sequence of constellation symbols. In eachsignaling interval (also referred to as a symbol period or time slot),signals 114 ₁ and 114 ₂ carry digital values that represent the in-phase(I) component and quadrature (Q) component, respectively, of acorresponding constellation symbol intended for transmission using afirst (e.g., X) polarization of light. Signals 114 ₃ and 114 ₄ similarlycarry digital values that represent the I and Q components,respectively, of the corresponding constellation symbol intended fortransmission using a second (e.g., Y) polarization of light.

E/O converter 140 operates to transform digital signals 114 ₁-114 ₄ intoa corresponding modulated optical output signal 142. More specifically,drive circuits 118 ₁ and 118 ₂ transform digital signals 114 ₁ and 114₂, as known in the art, into electrical analog drive signals I_(X) andQ_(X), respectively. Drive signals I_(X) and Q_(X) are then used, in aconventional manner, to drive an I-Q modulator 124 _(X). In response todrive signals I_(X) and Q_(X), I-Q modulator 124 _(X) operates tomodulate an X-polarized beam 122 _(X) of light supplied by a lasersource 120 as indicated in FIG. 12 , thereby generating a modulatedoptical signal 126 _(X).

Drive circuits 118 ₃ and 118 ₄ similarly transform digital signals 114 ₃and 114 ₄ into electrical analog drive signals I_(Y) and Q_(Y),respectively. In response to drive signals I_(Y) and Q_(Y), an I-Qmodulator 124 _(Y) operates to modulate a Y-polarized beam 122 _(Y) oflight supplied by laser source 120 as indicated in FIG. 12 , therebygenerating a modulated optical signal 126 _(Y). A polarization beamcombiner 128 operates to combine modulated optical signals 126 _(X) and126 _(Y), thereby generating optical output signal 142 (also see FIG. 1).

FIG. 13 shows a block diagram of optical receiver 108 that can be usedin system 100 (FIG. 1 ) according to an embodiment.

O/E converter 150 comprises an optical hybrid 159, light detectors 161₁-161 ₄, analog-to-digital converters (ADCs) 166 ₁-166 ₄, and an opticallocal-oscillator (OLO) source 156. Optical hybrid 159 has (i) two inputports labeled S and R and (ii) four output ports labeled 1 through 4.Input port S receives optical signal 142′ (also see FIG. 1 ). Input portR receives an OLO signal 158 generated by OLO source 156. OLO signal 158has an optical-carrier wavelength (frequency) that is sufficiently closeto that of signal 142′ to enable coherent (e.g., intradyne) detection ofthe latter signal. OLO signal 158 can be generated, e.g., using arelatively stable laser whose output wavelength (frequency) isapproximately the same as the carrier wavelength (frequency) of opticalsignal 142.

In an example embodiment, optical hybrid 159 operates to mix inputsignal 142′ and OLO signal 158 to generate different mixed (e.g., byinterference) optical signals (not explicitly shown in FIG. 13 ). Lightdetectors 161 ₁-161 ₄ then convert the mixed optical signals into fourelectrical signals 162 ₁-162 ₄ that are indicative of complex valuescorresponding to two orthogonal-polarization components of signal 142′.For example, electrical signals 162 ₁ and 162 ₂ may be an analog Isignal and an analog Q signal, respectively, corresponding to a first(e.g., horizontal, h) polarization component of signal 142′. Electricalsignals 162 ₃ and 162 ₄ may similarly be an analog I signal and ananalog Q signal, respectively, corresponding to a second (e.g.,vertical, v) polarization component of signal 142′. Note that theorientation of the h and v polarization axes at receiver 108 may notcoincide with the orientation of the X and Y polarization axes attransmitter 104.

Each of electrical signals 162 ₁-162 ₄ is converted into digital form ina corresponding one of ADCs 166 ₁-166 ₄. Optionally, each of electricalsignals 162 ₁-162 ₄ may be amplified in a corresponding electricalamplifier (not explicitly shown) prior to the resulting signal beingconverted into digital form. Digital signals 168 ₁-168 ₄ produced byADCs 166 ₁-166 ₄ are then processed by a DSP 170, which implements,inter alia, electronic decoder 160 (see FIG. 1 ).

In an example embodiment, in addition to the above-described decoding,DSP 170 may perform one or more of the following: (i) signal processingdirected at dispersion compensation; (ii) signal processing directed atcompensation of nonlinear distortions; (iii) electronic polarizationde-multiplexing; and (iv) FEC decoding.

According to an example embodiment disclosed above, e.g., in the summarysection and/or in reference to any one or any combination of some or allof FIGS. 1-13 , provided is an apparatus comprising: a digital encoder(e.g., 700, FIG. 7 ) having, at least, first and second digital stages(e.g., 610 ₁, 710 ₂, FIG. 7 ) to produce a stream of symbols (e.g., 712,FIG. 7 ) from a bitstream (e.g., 608 ₁, FIG. 7 ), the first digitalstage being configured to separately encode segments of a same number ofbits (e.g., k₁) of the bitstream into first sequences of symbols suchthat each of the first sequences has a same first length (e.g., n₁) anda respective total energy of the symbols therein lower than a threshold(e.g., E₁+δE), the second digital stage being configured to encodeblocks of the first sequences into second sequences of symbols, each ofthe blocks having a same number of first sequences therein, each of thesecond sequences having a same second length (e.g., n₂); and wherein atotal energy of symbols of any of the second sequences divided by thesecond length is smaller than the threshold divided by the first length.Herein, δE denotes an infinitesimal value, e.g., 0.01% of E₁.

In some embodiments of the above apparatus, the threshold is smallerthan an average energy (e.g., U_(n), Eq. (4)) of an unshaped sequence ofsymbols of the first length.

In some embodiments of any of the above apparatus, the second length isan integer multiple of the first length, the integer being two or more.

In some embodiments of any of the above apparatus, the first digitalstage is configured to produce the first sequences such that symbols oflarger energy are less common therein than symbols of lower energy.

In some embodiments of any of the above apparatus, the second stage isconfigured to produce the second sequences such that symbols of largerenergy are less common therein than symbols of lower energy.

In some embodiments of any of the above apparatus, the threshold issmaller than an average energy of the symbols times the first length,the average being taken over a corresponding constellation of thesymbols.

In some embodiments of any of the above apparatus, the digital encoderis configured to create the stream of symbols from the second sequencesby using some of the bits of the bitstream as signs for the symbols.

In some embodiments of any of the above apparatus, the apparatus furthercomprises an optical modulator (e.g., 124, FIG. 12 ) and electronicdriver thereof (e.g., 118, FIG. 12 ); and wherein the digital encoder isconfigured to cause the electronic driver to operate the opticalmodulator to produce a modulated optical carrier carrying the secondsequences of symbols.

In some embodiments of any of the above apparatus, the second length isan integer multiple of the first length, the integer being three ormore.

In some embodiments of any of the above apparatus, the second digitalstage comprises a constellation demapper (e.g., 620 ₂, FIG. 6 )connected to receive the first sequences of symbols.

According to another example embodiment disclosed above, e.g., in thesummary section and/or in reference to any one or any combination ofsome or all of FIGS. 1-13 , provided is an apparatus comprising anoptical data transmitter (e.g., 104, FIGS. 1, 12 ) that comprises anoptical front end (e.g., 140, FIGS. 1, 12 ) and a digital shapingencoder (e.g., 110, FIG. 1 ; 112, FIG. 12, 600 , FIG. 6 ; 700, FIG. 7 ;800, FIG. 8 ), the digital shaping encoder being configured to: encode abitstream (e.g., 608 ₁, FIGS. 6, 7, 8 ) into a stream (e.g., 632, FIG. 6; 732, FIGS. 7, 8 ) of symbols of a constellation; and cause the opticalfront end to produce a modulated optical signal (e.g., 142, FIG. 1 )carrying the stream of symbols; and wherein the shaping encoder isconfigured to generate a symbol sequence for the stream in response to asegment of the bitstream, the symbol sequence including an integernumber (e.g., ρ=n₂/n₁) of non-overlapping subsequences of a fixed length(e.g., n₁), any one of the subsequences having a transmit energy notexceeding a first threshold energy (e.g., E₁), the symbol sequencehaving a transmit energy not exceeding a second threshold energy (e.g.,E₂) and being generated such that some symbols of higher energy have alower probability of being generated by the digital shaping encoder thanother symbols of lower energy, the first threshold energy being smallerthan an average energy (e.g., U_(n), Eq. (4)) of an unshaped sequence ofthe fixed length of the symbols of the constellation, the secondthreshold energy being smaller than the integer number of the firstthreshold energies (e.g., Eq. (5)), the integer number being greaterthan one.

In some embodiments of the above apparatus, the digital shaping encodercomprises a first shaping stage (e.g., 602 ₁, FIG. 6 ) and at least asecond shaping stage (e.g., 602 ₂, FIG. 6 ), the first shaping stagebeing configured to generate a first amplitude stream (e.g., 612 ₁, FIG.6 ) in response to the bitstream, the first amplitude stream being astream of non-overlapping amplitude sequences of the fixed length, eachone of the amplitude sequences having an energy not exceeding the firstthreshold energy; and wherein the digital shaping encoder is configuredto generate the symbol sequence based on the first amplitude stream.

In some embodiments of any of the above apparatus, the second shapingstage comprises a constellation demapper (e.g., 620 ₂, FIG. 6 )connected to receive the first amplitude stream.

In some embodiments of any of the above apparatus, the second shapingstage is configured to generate a second amplitude stream (e.g., 612 ₂,FIG. 6 ) in response to the first amplitude stream, the second amplitudestream being a stream of non-overlapping amplitude sequences of a largersecond fixed length, each one of the amplitude sequences of said secondfixed length having an energy not exceeding the second threshold energy;and wherein the digital shaping encoder is configured to generate thesymbol sequence based on the second amplitude stream.

In some embodiments of any of the above apparatus, the shaping encodercomprises a memory (e.g., 820, FIG. 8 ) having stored therein a codebook(e.g., 900, generated in accordance with FIGS. 9A-9D) for convertingdifferent segments of the bitstream into corresponding differentamplitude sequences, each of the amplitude sequences having an energynot exceeding the second threshold energy and having a length equal to aproduct of the integer number and the fixed length.

In some embodiments of any of the above apparatus, the digital shapingencoder comprises a first shaping stage (e.g., 602 ₁, FIG. 6 ) and atleast a second shaping stage (e.g., 602 ₂, FIG. 6 ), the first shapingstage being nested in the second shaping stage, the first shaping stagebeing configured to apply a first shaping code to the segment of thebitstream, the second shaping stage being configured to apply a secondshaping code to the segment of the bitstream, the second shaping codehaving a smaller code rate than the first shaping code (e.g.,k/n₂<k/n₁).

In some embodiments of any of the above apparatus, the integer number isgreater than ten.

According to yet another example embodiment disclosed above, e.g., inthe summary section and/or in reference to any one or any combination ofsome or all of FIGS. 1-13 , provided is an apparatus comprising anoptical data transmitter (e.g., 104, FIGS. 1, 12 ) that comprises anoptical front end (e.g., 140, FIGS. 1, 12 ) and a digital signalprocessor (e.g., 110, FIG. 1 ; 112, FIG. 12 ), the digital signalprocessor being configured to: encode a bitstream (e.g., 608 ₁, FIGS. 6,7, 8 ) to generate a stream (e.g., 632, FIG. 6 ; 732, FIGS. 7, 8 ) ofsymbols of a constellation; and drive the optical front end to cause amodulated optical signal (e.g., 142, FIG. 1 ) generated by the opticalfront end to carry the symbols of the stream; and wherein the digitalsignal processor comprises a shaping encoder (e.g., 600, FIG. 6 ; 700,FIG. 7 ; 800, FIG. 8 ) configured to generate a symbol sequence for thestream in response to a segment of the bitstream, the symbol sequenceincluding an integer number (e.g., ρ=n₂/n₁) of non-overlappingsubsequences of a fixed length (e.g., n₁), any one of the subsequenceshaving a transmit energy not exceeding a first threshold energy (e.g.,E₁), the symbol sequence having a transmit energy not exceeding a secondthreshold energy (e.g., E₂) and being generated such that some symbolsof higher energy have a lower probability of being generated by theshaping encoder than other symbols of lower energy, the first thresholdenergy being smaller than an average energy (e.g., U_(n), Eq. (4)) of anunshaped sequence of the fixed length of the symbols of theconstellation, the second threshold energy being smaller than theinteger number of the first threshold energies (e.g., Eq. (5)), theinteger number being greater than one.

While this disclosure includes references to illustrative embodiments,this specification is not intended to be construed in a limiting sense.Various modifications of the described embodiments, as well as otherembodiments within the scope of the disclosure, which are apparent topersons skilled in the art to which the disclosure pertains are deemedto lie within the principle and scope of the disclosure, e.g., asexpressed in the following claims.

Some embodiments can be embodied in the form of methods and apparatusesfor practicing those methods. Some embodiments can also be embodied inthe form of program code recorded in tangible media, such as magneticrecording media, optical recording media, solid state memory, floppydiskettes, CD-ROMs, hard drives, or any other non-transitorymachine-readable storage medium, wherein, when the program code isloaded into and executed by a machine, such as a computer, the machinebecomes an apparatus for practicing the patented invention(s). Someembodiments can also be embodied in the form of program code, forexample, stored in a non-transitory machine-readable storage mediumincluding being loaded into and/or executed by a machine, wherein, whenthe program code is loaded into and executed by a machine, such as acomputer or a processor, the machine becomes an apparatus for practicingthe patented invention(s). When implemented on a general-purposeprocessor, the program code segments combine with the processor toprovide a unique device that operates analogously to specific logiccircuits.

Unless explicitly stated otherwise, each numerical value and rangeshould be interpreted as being approximate as if the word “about” or“approximately” preceded the value or range.

It will be further understood that various changes in the details,materials, and arrangements of the parts which have been described andillustrated in order to explain the nature of this disclosure may bemade by those skilled in the art without departing from the scope of thedisclosure, e.g., as expressed in the following claims.

The use of figure numbers and/or figure reference labels in the claimsis intended to identify one or more possible embodiments of the claimedsubject matter in order to facilitate the interpretation of the claims.Such use is not to be construed as necessarily limiting the scope ofthose claims to the embodiments shown in the corresponding figures.

Although the elements in the following method claims, if any, arerecited in a particular sequence with corresponding labeling, unless theclaim recitations otherwise imply a particular sequence for implementingsome or all of those elements, those elements are not necessarilyintended to be limited to being implemented in that particular sequence.

Reference herein to “one embodiment” or “an embodiment” means that aparticular feature, structure, or characteristic described in connectionwith the embodiment can be included in at least one embodiment of thedisclosure. The appearances of the phrase “in one embodiment” in variousplaces in the specification are not necessarily all referring to thesame embodiment, nor are separate or alternative embodiments necessarilymutually exclusive of other embodiments. The same applies to the term“implementation.”

Unless otherwise specified herein, the use of the ordinal adjectives“first,” “second,” “third,” etc., to refer to an object of a pluralityof like objects merely indicates that different instances of such likeobjects are being referred to, and is not intended to imply that thelike objects so referred-to have to be in a corresponding order orsequence, either temporally, spatially, in ranking, or in any othermanner.

Unless otherwise specified herein, in addition to its plain meaning, theconjunction “if” may also or alternatively be construed to mean “when”or “upon” or “in response to determining” or “in response to detecting,”which construal may depend on the corresponding specific context. Forexample, the phrase “if it is determined” or “if [a stated condition] isdetected” may be construed to mean “upon determining” or “in response todetermining” or “upon detecting [the stated condition or event]” or “inresponse to detecting [the stated condition or event].”

Also for purposes of this description, the terms “couple,” “coupling,”“coupled,” “connect,” “connecting,” or “connected” refer to any mannerknown in the art or later developed in which energy is allowed to betransferred between two or more elements, and the interposition of oneor more additional elements is contemplated, although not required.Conversely, the terms “directly coupled,” “directly connected,” etc.,imply the absence of such additional elements. The same type ofdistinction applies to the use of terms “attached” and “directlyattached,” as applied to a description of a physical structure. Forexample, a relatively thin layer of adhesive or other suitable bindercan be used to implement such “direct attachment” of the twocorresponding components in such physical structure.

The described embodiments are to be considered in all respects as onlyillustrative and not restrictive. In particular, the scope of thedisclosure is indicated by the appended claims rather than by thedescription and figures herein. All changes that come within the meaningand range of equivalency of the claims are to be embraced within theirscope.

A person of ordinary skill in the art would readily recognize that stepsof various above-described methods can be performed by programmedcomputers. Herein, some embodiments are intended to cover programstorage devices, e.g., digital data storage media, which are machine orcomputer readable and encode machine-executable or computer-executableprograms of instructions where said instructions perform some or all ofthe steps of methods described herein. The program storage devices maybe, e.g., digital memories, magnetic storage media such as a magneticdisks or tapes, hard drives, or optically readable digital data storagemedia. The embodiments are also intended to cover computers programmedto perform said steps of methods described herein.

The description and drawings merely illustrate the principles of thedisclosure. It will thus be appreciated that those of ordinary skill inthe art will be able to devise various arrangements that, although notexplicitly described or shown herein, embody the principles of thedisclosure and are included within its spirit and scope. Furthermore,all examples recited herein are principally intended expressly to beonly for pedagogical purposes to aid the reader in understanding theprinciples of the disclosure and the concepts contributed by theinventor(s) to furthering the art, and are to be construed as beingwithout limitation to such specifically recited examples and conditions.Moreover, all statements herein reciting principles, aspects, andembodiments of the disclosure, as well as specific examples thereof, areintended to encompass equivalents thereof.

The functions of the various elements shown in the figures, includingany functional blocks labeled as “processors” and/or “controllers,” maybe provided through the use of dedicated hardware as well as hardwarecapable of executing software in association with appropriate software.When provided by a processor, the functions may be provided by a singlededicated processor, by a single shared processor, or by a plurality ofindividual processors, some of which may be shared. Moreover, explicituse of the term “processor” or “controller” should not be construed torefer exclusively to hardware capable of executing software, and mayimplicitly include, without limitation, digital signal processor (DSP)hardware, network processor, application specific integrated circuit(ASIC), field programmable gate array (FPGA), read only memory (ROM) forstoring software, random access memory (RAM), and non volatile storage.Other hardware, conventional and/or custom, may also be included.Similarly, any switches shown in the figures are conceptual only. Theirfunction may be carried out through the operation of program logic,through dedicated logic, through the interaction of program control anddedicated logic, or even manually, the particular technique beingselectable by the implementer as more specifically understood from thecontext.

As used in this application, the term “circuitry” may refer to one ormore or all of the following: (a) hardware-only circuit implementations(such as implementations in only analog and/or digital circuitry); (b)combinations of hardware circuits and software, such as (as applicable):(i) a combination of analog and/or digital hardware circuit(s) withsoftware/firmware and (ii) any portions of hardware processor(s) withsoftware (including digital signal processor(s)), software, andmemory(ies) that work together to cause an apparatus, such as a mobilephone or server, to perform various functions); and (c) hardwarecircuit(s) and or processor(s), such as a microprocessor(s) or a portionof a microprocessor(s), that requires software (e.g., firmware) foroperation, but the software may not be present when it is not needed foroperation.” This definition of circuitry applies to all uses of thisterm in this application, including in any claims. As a further example,as used in this application, the term circuitry also covers animplementation of merely a hardware circuit or processor (or multipleprocessors) or portion of a hardware circuit or processor and its (ortheir) accompanying software and/or firmware. The term circuitry alsocovers, for example and if applicable to the particular claim element, abaseband integrated circuit or processor integrated circuit for a mobiledevice or a similar integrated circuit in server, a cellular networkdevice, or other computing or network device.

It should be appreciated by those of ordinary skill in the art that anyblock diagrams herein represent conceptual views of illustrativecircuitry embodying the principles of the disclosure. Similarly, it willbe appreciated that any flow charts, flow diagrams, state transitiondiagrams, pseudo code, and the like represent various processes whichmay be substantially represented in computer readable medium and soexecuted by a computer or processor, whether or not such computer orprocessor is explicitly shown.

What is claimed is:
 1. An apparatus, comprising: a digital encoderhaving, at least, first and second digital stages to produce a stream ofsymbols from a bitstream, the first digital stage being configured toseparately encode segments of a same number of bits of the bitstreaminto first sequences of symbols such that each of the first sequenceshas a same first length and a respective total energy of the symbolstherein lower than a threshold, the second digital stage beingconfigured to encode blocks of the first sequences into second sequencesof symbols, each of the blocks having a same number of the firstsequences therein, each of the second sequences having a same secondlength; and wherein a total energy of the symbols of any of the secondsequences divided by the second length is smaller than the thresholddivided by the first length; and wherein the second digital stagecomprises a constellation demapper connected to receive the firstsequences of symbols.
 2. The apparatus of claim 1, wherein the thresholdis smaller than an average energy of an unshaped sequence of symbols ofthe first length.
 3. The apparatus of claim 1, wherein the second lengthis an integer multiple of the first length, the integer being two ormore.
 4. The apparatus of claim 1, wherein the first digital stage isconfigured to produce the first sequences such that symbols of largerenergy are less common therein than symbols of lower energy.
 5. Theapparatus of claim 4, wherein the second digital stage is configured toproduce the second sequences such that symbols of larger energy are lesscommon therein than symbols of lower energy.
 6. The apparatus of claim4, further comprising an optical modulator and electronic driverthereof; and wherein the digital encoder is configured to cause theelectronic driver to operate the optical modulator to produce amodulated optical carrier carrying the second sequences of symbols. 7.The apparatus of claim 1, wherein the threshold is smaller than anaverage energy of the symbols times the first length, the average beingtaken over a corresponding constellation of the symbols.
 8. Theapparatus of claim 7, further comprising an optical modulator andelectronic driver thereof; and wherein the digital encoder is configuredto cause the electronic driver to operate the optical modulator toproduce a modulated optical carrier carrying the second sequences ofsymbols.
 9. The apparatus of claim 1, wherein the digital encoder isconfigured to create the stream of symbols from the second sequences ofsymbols by using some of the bits of the bitstream as signs for thesymbols.
 10. The apparatus of claim 9, further comprising an opticalmodulator and an electronic driver thereof; and wherein the digitalencoder is configured to cause the electronic driver to operate theoptical modulator to produce a modulated optical carrier carrying thestream of symbols.
 11. The apparatus of claim 1, further comprising anoptical modulator and electronic driver thereof; and wherein the digitalencoder is configured to cause the electronic driver to operate theoptical modulator to produce a modulated optical carrier carrying thesecond sequences of symbols.
 12. The apparatus of claim 1, wherein thesecond length is an integer multiple of the first length, the integerbeing three or more.
 13. An apparatus comprising an optical datatransmitter that comprises an optical front end and a digital shapingencoder, the digital shaping encoder being configured to: encode abitstream into a stream of symbols of a constellation; and cause theoptical front end to produce a modulated optical signal carrying thestream of symbols; and wherein the digital shaping encoder is configuredto generate a symbol sequence for the stream in response to a segment ofthe bitstream, the symbol sequence including an integer number ofnon-overlapping subsequences of a fixed length, any one of thesubsequences having a transmit energy not exceeding a first thresholdenergy, the symbol sequence having a transmit energy not exceeding asecond threshold energy and being generated such that some symbols ofhigher energy have a lower probability of being generated by the digitalshaping encoder than other symbols of lower energy, the first thresholdenergy being smaller than an average energy of an unshaped sequence ofthe fixed length of the symbols of the constellation, the secondthreshold energy being smaller than the integer number of the firstthreshold energies, the integer number being greater than one.
 14. Theapparatus of claim 13, wherein the digital shaping encoder comprises afirst shaping stage and at least a second shaping stage, the firstshaping stage being configured to generate a first amplitude stream inresponse to the bitstream, the first amplitude stream being a stream ofnon-overlapping amplitude sequences of the fixed length, each one of theamplitude sequences having an energy not exceeding the first thresholdenergy; and wherein the digital shaping encoder is configured togenerate the symbol sequence based on the first amplitude stream. 15.The apparatus of claim 14, wherein the second shaping stage comprises aconstellation demapper connected to receive the first amplitude stream.16. The apparatus of claim 14, wherein the second shaping stage isconfigured to generate a second amplitude stream in response to thefirst amplitude stream, the second amplitude stream being a stream ofnon-overlapping amplitude sequences of a larger second fixed length,each one of the amplitude sequences of said second fixed length havingan energy not exceeding the second threshold energy; and wherein thedigital shaping encoder is configured to generate the symbol sequencebased on the second amplitude stream.
 17. The apparatus of claim 13,wherein the shaping encoder comprises a memory having stored therein acodebook for converting different segments of the bitstream intocorresponding different amplitude sequences, each of the amplitudesequences having an energy not exceeding the second threshold energy andhaving a length equal to a product of the integer number and the fixedlength.
 18. The apparatus of claim 13, wherein the digital shapingencoder comprises a first shaping stage and at least a second shapingstage, the first shaping stage being nested in the second shaping stage,the first shaping stage being configured to apply a first shaping codeto the segment of the bitstream, the second shaping stage beingconfigured to apply a second shaping code to the segment of thebitstream, the second shaping code having a smaller code rate than thefirst shaping code.
 19. The apparatus of claim 13, wherein the integernumber is greater than ten.
 20. An apparatus, comprising: a digitalencoder having, at least, first and second digital stages to produce astream of symbols from a bitstream, the first digital stage beingconfigured to separately encode segments of a same number of bits of thebitstream into first sequences of symbols such that each of the firstsequences has a same first length and a respective total energy of thesymbols therein lower than a threshold, the second digital stage beingconfigured to encode blocks of the first sequences into second sequencesof symbols, each of the blocks having a same number of the firstsequences therein, each of the second sequences having a same secondlength; wherein a total energy of the symbols of any of the secondsequences divided by the second length is smaller than the thresholddivided by the first length; and wherein the threshold is smaller thanat least one of (i) an average energy of the symbols times the firstlength where the average is taken over a corresponding constellation ofthe symbols and (ii) an average energy of an unshaped sequence ofsymbols of the first length.
 21. The apparatus of claim 20, wherein thethreshold is smaller than an average energy of an unshaped sequence ofsymbols of the first length.
 22. The apparatus of claim 20, wherein thethreshold is smaller than an average energy of the symbols times thefirst length, the average being taken over a corresponding constellationof the symbols.
 23. The apparatus of claim 20, further comprising anoptical modulator and electronic driver thereof; and wherein the digitalencoder is configured to cause the electronic driver to operate theoptical modulator to produce a modulated optical carrier carrying thesecond sequences of symbols.